A friend who's an avid hiker recommends Wilder Ranch State Park, but I haven't been there.
My favorite beach walk with dramatic cliffs is at the Seascape Resort hotel in Aptos. The hotel is legally required to allow public access to their beach, and they do so completely willingly, no sneaking around required. You don't have to be sheepish about using their nice bathroom or buying food or drink, even if you're dressed like a hiker. There is no public right to park in the hotel's outdoor parking lot, but there is no gate, guards or fee and never a shortage of space.
Directions:
Take Hwy 1 south from Santa Cruz.
Exit San Andreas Rd 0.8 mi.
R at Seascape Blvd 0.8 mi.
Continue on Seascape Resort Dr 280 ft.
Turn left to stay on Seascape Resort Dr.
Destination is on the right 260 ft.